We studied the group structure of Kac-Moody groups and associated K-groups. Especially we are interested in the Kac-Moody groups over Z.We selected several types of Kac-Moody groups and studied their relations. In particular, we determined the group structure of their abelian quotients. Also, we studied the group structure of K-group K_2 (2, Z [1/p]). One of our results is as follows. For a prime number p<greater than or equal>5, we found that K_2 (2, Z [1/p]) is not central, and furthermore K_2 (2, Z [1/p]) *Z_* x Z_<p-1>. To obtain these results, we use several braid relations and meta-abelianizations. The main tool is the so-called Dennis-Stein symbol. If p=6k+1, k=2^lm, (2, m) =1, then we choose the Dennis-Stein symbol named d (-2^<l+1>,3m). If p=6k-1, k=2^lm, (2, m), then we choose the Dennis-Stein symbol named d (2^<l+1>,3m). In the meta-abelian quotient of St (2, Z [1/p]), these Dennis-Stein symbols work well, which gave the non-centrality of K_2 (2, Z [1/p]) and so on. And, we
… Morestudied the relation between alternating groups and hyperbolic Coxeter groups. Then, we constructed a compact Riemann Surface of genus 1+ {(p-1) ! (p-6) /24} for every prime number p<greater than or equal>11. As well as these results, we studied several related topics. For example, a research of quantum groups, a research of algebraic geometry, a research of finite groups, a research of Lie groups, a research of number theory, a research of representation theory of rings, etc. And we announced these results in several international conferences, and published these results in several international academic journals. Also we are prepairing to publish some new results in certain international academic journals.・一般のKac-Moody群において、大域Gauss分解が成立していることが発見された。これは、共役類に関しての局所Gauss分解への道が大いに有望であることを示しており、大変に画期的なことである。・代数的K群のうちで、pが5以上の素数に対するものして、K_2(2,Z[1/p])が中心的ではないこと、さらにK_2(2,Z[1/p])〓Z_∞×Z_<P-1>となっていることが示された。これらは、その構造を具体的に述べたもので、今までにない新しい結果である。・交代群と双曲型Coxeter群との間の新たな関係を発見し、それを用いて種数が1+{(p-1)!(p-6)/24}となるコンパクトRiemann面を構成した。さらに、その応用として、SL(2,Z)の合同部分群と(].SU.[)の正規閉包との間の構造について研究した。・以上の研究と共に、量子群論的、代数幾何学的、有限群論的、リー群論的、数論的、環の表現論的考察を研究分担者と行った。そして、得られた研究成果について国際研究集会や学術雑誌等で発表した。 Less